Albrecht durer ppt

Albrecht Dürer: Art, Geometry, and Printing

"Without [the art of measurement] no one

can become a true artisan."

Albrecht DürerThe Painter's Manual

(1525)

Welcome to the Renaissance!

Vasari's Lives of the Artists: "Rebirth" of classical forms.

But: What are classical forms?

Medieval or Classical?

Notre Dame, Paris Crude, barbaric.

("Gothic", but actually, medieval)

Pantheon, Rome Domes, columns

Medieval or Classical?

Book of Hours (Duke of Berry)

Non-realistic

Boscoreale (Naples) Realistic depiction of

space

Medieval or Classical?

Merovingian minuscule.

"Form of torture"

Latin capitals. Easy to read.

A Quest for Lost Arts

How did the Romans and Greeks make their art?

Many artifacts, but few documents.

Reverse engineering!

Raphael's School of Athens (1511)

A Clasical Education

"Libera" (free men) vs. "Manus" (hand work) Plato's educational system (The Republic)

Arithmetic (number properties) Geometry (Euclidean) Music (as applied arithmetic) Astronomy (as applied geometry)

Math is all the liberal arts

The Degeneration of Philosophy

Martianus Capella (fl. 420) The Marriage of Philo and Mercury Geometry reduced to areas, lengths, geography

Boethius (fl. 500) Quadrivium Basis for medieval education

The Rebirth of Western Mathematics

Islamic Civilization Preserved and extended Greek geometry Abu'l-Wafa On the Parts of Geometry Necessary for

Artisans Geometric designs

The Spanish Connection Campanus of Novara (fl. 1300)

New edition of Elements Geometry more than lengths and areas

The Mathematization of Art

Cathedral Building Jordanus Nemorarius

(fl. 1225): On Weights Filippo Brunelleschi

(1377-1446) Architect of duomo Perspective Art is applied math!

Leon Battista Alberti (1404-1472)

On Painting (1435) Painting as a liberal art Mathematical basis

On Architecture (1452) Study of Roman ruins,

Vitruvius Importance of ratios and

proportionSanta Maria Novella (Florence)

Vitruvius Ten Books on Architecture

Inspired Alberti, Pacioli, da Vinci

Importance of ratios Face: Body = 1:10 Arms: Legs = 1:1 Foot: Body = 1:6 Forearm: Body = 1:4

Aesthetics quantifiable!

da Vinci's "Vitruvian Man"

Every Artist a Geometer...

Piero della Francesca (1415-1492) "Flagellation" one of

oldest to use true perspective.

Treatise on the Abacus Five Books on the

Regular Solids.

...and every Geometer an Artist

Luca Pacioli (1446-1517) Summary of

Arithmetic, Geometry, Proportion, and Proportionality (1494)

The Divine Proportion (1509), includes Francesca's Five Books (illustrated by da Vinci)

Nürnberg in the Fifteenth Century

Political center (Diet meetingplace)

Oldest paper mills in Germany

Printing center (Regiomonantus's press)

Weltchronik of Wolgemut

Albrecht Dürer

Born Nürnberg, 1471 Education of an Artist

Apprenticed to Wolgemut Carved wood blocks for print illustrations Two Italian trips (1494-5, 1505-7) Met Pacioli, studied Alberti, Euclid, Vitruvius

The Painter's Manual (1525)

"Nothing is more annoying to men of understand in than a blunder in a painting..."

Practical geometric construction for artists.

Theoretical geometry

Artist's Problem #1

How do you draw an object?

Composition of basic shapes.

Straight lines and circles...

...but other curves (e.g., helix) required for "realism."

Pope Benedict in Munich

Basic Drawing

Straight lines and circles easy to construct.

Paraphernalia of daily life often includes other shapes.

Crozier: Spiral and circular arcs.

Croziers from Metropolitan Mustum of Art

Dürer's Spirals

Divide AB into fourths at C, D, E.

Bisect D, F. Draw semicircles AB,

BC, CE, ED with centers alternating D, F (final semicircle DF with center at midpoint).

Non-Archimedean spiral.

Plan and Elevation

Representing space curves.

Dual viewpoint. Detailed instructions

on how to construct. Adapted from

architectural plans.

Conic Sections for Artists

The ellipse as a conic section.

Dürer invents German name ("egg-shaped line")

Similar constructions for parabola ("burning line") and hyperbola ("fork shaped line").

First contour maps?

Dürer's Polygons

Simple construction for regular hexagon: no need to change compass opening.

Join alternate vertices for equilateral triangle.

The Heptagon

Side of heptagon is half side of equilateral triangle.

Method given by Abu'l-Wafa (On the Parts of Geometry Necessary for Artisans)

Dürer's method new to Europe?

Kepler: "Obviously" too short.

How Close to Exact?

H is vertex of regular heptagon.

For circle with diameter 20 feet, approximation gives side of heptagon as 8 feet, 7.92 inches vs. actual side of 8 feet, 8.13 inches.

Pentagons

Bisect radius OA at C. Join CB, and mark CD equal to CB. BD is side of pentagon.

Method given by Ptolemy in Almgest

Exact construction.

Rusty Compass Pentagon

AB as side, draw circleswith centers A, B and radius AB.

Join CD. Draw circle with center D and radius AB.

Draw EG to H and FG to I. Make IJ, HJ equal to AB.

ABHJI is equilateralpentagon.

Also Approximate

Pentagon is equilateral but not equiangular

Method from Geometria Deutsch of Matthias Roritzer

"Rusty compass" problems appear in Islamic geometry (Abu'l-Wafa gave exact construction for pentagon).

Tiling Patterns

Floor covering using polygons

Regular tiling using squares, triangles, octagons.

Heptagonal Tilings Pentagonal Tilings

A Near Miss...

Pentagons Joined rosettes Fill space with other figures

Kepler (1619) Critiqued Dürer's heptagon No mention of Dürer tilings

Penrose (1974)

...Or Not

Problem: How do you represent three-dimensional objects schematically?

"Framework" to show structure (Leonardo's drawing of rhombi-cuboctahedron from Pacioli's Divine Proportion)

Dürer's Polyhedral Nets

Dodecahedron Rhombicuboctahedron

Mapping the World

Can "unwrap" sphere in same way (Dürer).

Spherical projection problem.

Waldseemüller map (1506) first to use designation "America."

Artist's Problem #2

How do you make a painting look like a real scene?

More distant objects look smaller (foreshortening).

Convergence of parallel lines (perspective).

A Building in Nürnberg

Foreshortening

Used to convey sense of distance.

Dürer used (St. Jerome woodcut)

Mathematical basis: Objects of equal size subtend smaller angles at greater distances.

Practical Design

Inscriptions on buildings and columns.

Equal arcs make for equal apparent size (from vantage point).

Modern street writings.

Perspective

Radial symmetry about vanishing point.

Not in Classical Art! Renaissance

invention.

Woodcuts and Paintings

Dürer learns "secret" during 1506 visit to Italy.

Explains technique in Painter's Manual.

Ratios of Gold...

Pacioli's Divine Proportion Ancients constructed

temples and art using key ratios.

Ratio between diagonal of pentagon and side is "Golden Ratio"

...and Dross

More natural to use Fibonacci sequence

ALL Vitruvian ratios rational.

NO evidence for deliberate use of side/diagonal of pentagon.

Pacioli doesn't use golden ratio as basis.

A Brief History of Typography

Roman majuscule (CAPITAL LETTERS). Merovingian minuscule (lower case letters,

joined with ligatures). t, e, i, r: er et te ti tr

Merovingian script, from Luxeuil manuscript.(From http://medievalwriting.50megs.com/scripts/examples/luxeuil.htm)

Carolingian Minuscule

Alcuin and Charlemagne Reform of Merovingian

script Basis for modern Roman

lower case Far easier to read and

write! Degeneration of Roman

type?

Manuscripts and Printers

Printing press China/Korea Print money, playing cards.

Gutenberg What should letters look

like? Challenge: Make an "A" using

the Golden Ratio (or Fibonacci Rectangle)

Classic Fonts

Felice Feliciano Pacioli

Letters in appendix to Divine Proportion

Based on circle and square

Thick limbs = 1/9 height, thin limbs = 1/2 of thick

NO golden ratio.

Dürer's Letters

Circle and square Ratios

Square into four parts Thick limbs = 1/10

height, thin limbs 1/3 of thick.

Top center of width. Base serifs use circle

with radius 1/7 of side of square

Gothic Letters, Old Style

Squares atop squares 1/3 division of sides Vertical diagonals Letters as

components of other letters ("i" as basis for u, b, etc.)

Gothic Letters, New Style

Old style, with additional serifs, squares.

Roman type only upper case letters.

Dürer first (?) to geometrize lower case letters.

Reactions

Aesthetics Cresci (1560): Curves of actual letters not circles

and straight lines. Letters "ugly"

Practicality No one learns to write this way! Difficult to cut type metal this

precisely Impression printing has no need

for mathematical defined fonts.

The Computer Connection

Bit map fonts Graininess problem One image for each size of each letter

True-Type fonts Letters defined by lines and curves Curves beyond circle (splines)

Printers (and computer screens) Ink-jet: True-Type to bitmap to print Laser printer: True-Type to laser pointer to print

Some Final Thoughts

Thinking componentwise and choosing what is important are essential skills in the creation of art...and mathematics.

The constructive arts are an underutilized resource in the teaching of mathematics.